Bell ’ s Inequality Should Be Reconsidered in Quantum Language

Bell’s inequality itself is usually considered to belong to mathematics and not quantum mechanics. We think that this is making our understanding of Bell’ theory be confused. Thus in this paper, contrary to Bell’s spirit (which inherits Einstein’s spirit), we try to discuss Bell’s inequality in the framework of quantum theory with the linguistic Copenhagen interpretation. And we clarify that the violation of Bell’s inequality (i.e., whether or not Bell’s inequality holds) does not depend on whether classical systems or quantum systems, but depend on whether a combined measurement exists or not. And further we conclude that our argument (based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell’s philosophical argument (based on Einstein’s spirit).


Review: Quantum Language (=Measurement Theory (=MT)) 1.Introduction
Recently (cf.refs.[1]- [10], also see (B 0 ) -(B 3 ) later), we proposed quantum language, which was not only characterized as the metaphysical and linguistic turn of quantum mechanics but also the linguistic turn of dualistic idealism.And further we believe that quantum language should be regarded as the foundations of quantum information science.Quantum language is formulated as follows.
(A) quantum mechanical world view.That is, we think that the location of quantum language in the history of world-descriptions is as follows.
(B 1 ) to clarify the Copenhagen interpretation of quantum mechanics (cf.⑦ in Figure 1, refs.[2] [7] [11]), that is, the linguistic Copenhagen interpretation is the true figure of so-called Copenhagen interpretation.
In Bohr-Einstein debates (refs.[13] [14]), Einstein's standing-point (that is, "the moon is there whether one looks at it or not" (i.e., physics holds without observers)) is on the side of the realistic world view in Figure 1.On the other hand, we think that Bohr's standing point (that is, "to be is to be perceived" (i.e., there is no science without measurements)) is on the side of the linguistic world view in Figure 1 (though N. Bohr might believe that the Copenhagen interpretation (proposed by his school) belongs to physics).
In this paper, contrary to Bell's spirit (which inherits Einstein's spirit), we try to discuss Bell's inequality (refs.[15] [16] [17] [18]) in quantum language (i.e., quantum theory with the linguistic Copenhagen interpretation).And we clarify that whether or not Bell's inequality holds does not depend on whether classical systems or quantum systems (in Section 3), but depend on whether a combined measurement exists or not (in Section 2).And further we assert that our argument (based on the linguistic Copenhagen interpretation) should be regarded as a scientific representation of Bell's philosophical argument (based on Einstein's spirit).
Figure 1.The history of the world-descriptions.

Quantum Language (=Measurement Theory); Mathematical Preparations
Now we shall explain the measurement theory (A).
Consider an operator algebra ( ) B H (i.e., an operator algebra composed of all bounded linear operators on a Hilbert space H with the norm ( ) ), and consider the pair [ ] ( ) , called a basic structure.Here, ( ) ( ) is a C * -algebra, and  ( ( ) ) is a particular C * -algebra (called a W * -algebra) such that  is the weak closure of  in ( ) The measurement theory (=quantum language) is classified as follows. (C) measurement theory A C : quantum system theory when C : classical system theory when , the C * -algebra composed of all compact operators on a Hilbert space H, the (C 1 ) is called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics.Also, when  is commutative (that is, when  is characterized by ( ) 0 C Ω , the C * -algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space Ω (cf.[19] [20])), the (C 2 ) is called classical measurement theory (or, classical system theory).Also, note (cf.[19]) that, when ( ) Also, when ( ) "the space of all signed measures on Ω ", ( , where ν is some measure on Ω (cf.[19]).Also, the ( ) which is called a state space.It is well known (cf.[19]) that can be also identified with Ω (called a spectrum space or simply spectrum) such as For instance, in the above 2) we must clarify the meaning of the "value" of ( ) And the value of ( ) 0 F ρ is defined by the α .According to the noted idea (cf.[21]), an observable is defined as follows: 1 where 0 and I is the 0-element and the identity in  respectively. (c): for any countable decomposition { } , , , , holds that ( ) ( ) in the sense of weak * topology in  .
Remark 1.Quantum language has two formulations (i.e., the C * -algebraic formulation and the W * -algebraic formulation).In this paper, we devote ourselves to the W * -algebraic formulation, which may, from the mathematical point of view, be superiority to the C * -algebraic formulation.That is, in the above 2), the countable additivity (i.e., ( ) ( ) ) is naturally discussed in the W * -algebraic formulation.However, the C * -algebraic formulation has a merit such that we can use it without sufficient mathematical preparation.For the C * -algebraic version of this paper, see my preprint [10].

Axiom 1 [Measurement] and Axiom 2 [Causality]
. In addition to the above 1) and 2), in this paper we assume that ( ) Φ is said to be deterministic.If it is not deterministic, it is said to be non-deterministic or decoherence.Here note that, for any observable ( ) ≤ .The causality is represented by a Markov operator , :

The Linguistic Interpretation (=The Manual to Use Axioms 1 and 2)
In the above, Axioms 1 and 2 are kinds of spells, (i.e., incantation, magic words, metaphysical statements), and thus, it is nonsense to verify them experimentally.
Therefore, what we should do is not "to understand" but "to use".After learning Axioms 1 and 2 by rote, we have to improve how to use them through trial and error.
We can do well even if we do not know the linguistic interpretation.However, it is better to know the linguistic interpretation (=the manual to use Axioms 1 and 2), if we would like to make progress quantum language early.
The essence of the manual is as follows: (D) Only one measurement is permitted.And thus, the state after a measurement is meaningless since it cannot be measured any longer.Thus, the collapse of the wavefunction is prohibited (cf.[7]).We are not concerned with anything after measurement.That is, any statement including the phrase after the measurement is wrong.Also, the causality should be assumed only in the side of sys- tem, however, a state never moves.Thus, the Heisenberg picture should be adopted, and thus, the Schrödinger picture should be prohibited.Also, it is added that there is no probability without a measurement.and so on.For details, see [8]. , ×   be the product measurable space, i.e., the product space and the product σ-field   , which is defined by the smallest σ-field that contains a family Also, the measurement Note that the existence and the uniqueness of a generalized simultaneous observable ( ) in  are not assured in general, however the simultaneous observable always exists if observables O k ( ) . We consider the spatial tensor W * -algebra ( ) ( ) , and consider the product measurable space ( ) , 1, 2, , .
And let (which is also denoted by . Consider four observables: ( ) ( ) The four observables are said to be combinable if there exists an observable , for any ( ) . Also, the measurement ( ) ) does not hold in quantum systems but in classical systems (cf.ref. [8]).A certain combined observable plays an important role in the proof of the classical syllogism (cf.ref. [12]).
The following theorem is all of our insistence concerning Bell's inequality.We assert that this is the true Bell's inequality. ) [ ] ( ) . Define four correlation functions ( )  And further, we have Bell's inequality in quantum language as follows.
As the corollary of this theorem, we have the followings: Corollary 7. Consider the parallel measurement Here, note that the law of large numbers says: for sufficiently large N, ( ) Then, it holds, by the Formula (5), that , , , : ≡ − be a measurable functions.Define the correlation functions . Then, the following mathematical Bell's inequality (or precisely, CHSH inequality (cf.ref. [16])) holds: (E) This is easily proved as "the left-hand side of the above 8 This completes the proof.
Recall Theorem 6 (Bell's inequality in quantum language), in which we have, by the combinable condition, the probability space Therefore the proof of Theorem 6 and the above proof (E) are, from the mathematical point of view, the same.

"Bell's Inequality" Is Violated in Classical Systems as Well as Quantum Systems
In the previous section, we show that Theorem 6 (or Corollary 7) says (F 1 ) Under the combinable condition (cf.Definition 4), Bell's Inequality (5) (or, (7)) holds in both classical systems and quantum systems.
Or, equivalently, (F 2 ) If Bell's Inequality (5) (or ( 7)) is violated, then the combined observable does not exist, and thus, we cannot obtain the measured value (by the combined measurement).This is similar to the following elementary statement in quantum mechanics: ( 2 F′ ) We have no (generalized) simultaneous measurement of the position observable Q and the momentum observable P, and thus we cannot obtain the measured value (by the generalized simultaneous measurement), which may be, from Einstein's point of view, represented that "true value (or, hidden variable) of the position and momentum" does not exist.Since the error ∆ is usually defined by rough measured value true value ∆ = − , it is not easy to define the errors Q ∆ and P ∆ in Heisenberg's uncertainty principle This definition was completed and Heisenberg's uncertainty principle was proved in ref. [11].Also, according to the maxim of dualism: "To be is to be perceived" due to G. Berkeley, we think that it is not necessary to name that does not exist (or equivalently, that is not measured).
The above statement (F 2 ) makes us expect that (G) Bell's inequality (5) (or (7)) is violated in classical systems as well as quan- This (G) was already shown in my previous paper [2].However, I received a lot of questions concerning (G) from the readers.Thus, in this section, we again explain the (G) precisely.

Bell Test Experiment
In order to show the (G), three steps ( [ , , The correlation ( ) = i j is defined as follows: ( , , Now we have the following problem: (H) Find a measurement ( ) ( ) which is the same as the condition in Remark 5.
Let us answer this problem (H) in the two cases (i.e., classical case and quantum case), that is, , 1) the case of quantum system: ( ) ( ) in ( ) ( ) , where it should be noted that Further define the singlet state 0 ( ) 2 s e e e e ψ = ⊗ − ⊗ Thus we have the measurement . The followings are clear: for each ( ) { } ( ) For example, we easily see: ( ) a a a a a a a Therefore, the measurement satisfies the condition (H).
Then, it is clear that the measurement 2)' the case of classical systems: ( ) ( ) ( ) It is easy to show a lot of different answers from the above 2).For example, as a slight generalization of ( 9), define the probability measure ( ) ∈Ω × Ω .And assume that ( ) O : , , Thus, we have four observables L ∞ Ω × Ω (though the variables are not separable (cf. the Formula (13)).Then, it is clear that the measurement As defined by (9), consider four complex numbers ( ) ( ) we see, by (10), that ( ) ( ) ( ) ( ) Further, assume that the measured value is ( ) Then, the law of large numbers says that ( ) This and the Formula (18) say that Therefore, Bell's Inequality (5) (or ( 7)) is violated in classical systems as well as quantum systems.
which is similar as in Remark 5; 1) or in (H).

Conclusions
In Bohr-Einstein debates (refs.[13] [14]), Einstein's standing-point (that is, "the moon is there whether one looks at it or not" (i.e., physics holds without observ- ers)) is on the side of the realistic world view in Figure 1.On the other hand, we think that Bohr's standing point (that is, "to be is to be perceived" (i.e., there is no science without measurements)) is on the side of the linguistic world view in Figure 1.
In this paper, contrary to Bell's spirit (which inherits Einstein's spirit), we try to discuss Bell's inequality in Bohr's spirit (i.e., in the framework of quantum lan- It should be note that the concept of "hidden variable" is independent of measurements, thus, the (I 2 ) is a philosophical statement in Einstein's spirit, on the other hand, the (I 1 ) is a statement in Bohr's spirit (i.e., there is no science without measurements).It is sure that Bell's answer (I 2 ) is attractive philosophically, however, we believe in the scientific superiority of our answer (I 1 ).That is, we think that our (I 1 ) is a scientific representation of the philosophical (I 2 ).If so, we can, for the first time, understand Bell's inequality in science.That is, Theorem 6 is the true Bell's inequality.And we conclude that whether or not Bell's inequality holds does not depend on whether classical systems or quantum systems (in Section 3), but depend on whether the combined measurement exists or not (in Section 2).
We hope that our proposal will be examined from various points of view 1 .
Note that this theory (A) is not physics but a kind of language based on the S. Ishikawa DOI: 10.4236/jqis.2017.74011141 Journal of Quantum Information Science algebra, and let *  be the dual Banach space of  .That is, *  = {ρ | ρ is a continuous linear functional on  }, and the norm

Remark 9 .L
For completeness, note that the observables O ∞ Ω × Ω are not combinable in spite that these commute.Also, note that the Formulas (16) and(17) imply that .4236/jqis.2017.74011153 Journal of Quantum Information Science guage).And we show Theorem 6 (Bell's inequality in quantum language), which says the statement (F 2 ), that is, (I 1 ) (≡(F 2 )): If Bell's Inequality (5) (or (7)) is violated, then the combined observable does not exist, and thus, we cannot obtain the measured value (by the measurement of the combined observable).Also, recall that Bell's original argument says, roughly speaking, that (I 2 ) If the mathematical Bell's Inequality (8) is violated in Bell test experiment (the quantum case of Section 3.1), then hidden variables do not exist.

Bell's Inequality Always Holds in Classical and Quantum Systems Our Main Assertion about Bell's Inequality
S. Ishikawa DOI: 10.4236/jqis.2017.74011146 Journal of Quantum Information Science 2.